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Journal of Convex Analysis 27 (2020), No. 3, 811--832
Copyright Heldermann Verlag 2020

Differentiability of the Argmin Function and a Minimum Principle for Semiconcave Subsolutions

Julius Ross
Dept. of Mathematics, Statistics and Computer Science, University of Illinois, Chicago, IL 60607, U.S.A.

David Witt Nyström
Dept. of Mathematical Sciences, University of Gothenburg, 41296 Göteborg, Sweden


Suppose $f(x,y) + \frac{\kappa}{2} \|x\|^2 - \frac{\sigma}{2}\|y\|^2$ is convex where $\kappa\ge 0, \sigma>0$, and the argmin function $\gamma(x) = \{ \gamma : \inf_y f(x,y) = f(x,\gamma)\}$ exists and is single valued. We will prove $\gamma$ is differentiable almost everywhere. As an application we deduce a minimum principle for certain semiconcave subsolutions.

Keywords: Argmin function, differentiability, minimum principle, semiconcave subsolutions.

MSC: 28B20, 58C06.

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